# How do you find the area of the surface generated by rotating the curve about the y-axis y=1/4x^4+1/8x^-2, 1<=x<=2?

Apr 13, 2017

$\frac{253 \pi}{20}$

#### Explanation:

By Power Rule,

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{3} - {x}^{- 3} / 4$

So, the arc length element is:

sqrt{1+(dy/dx)^2}=sqrt(1+(x^3)^2-1/2+(x^(-3)/4)^2) =sqrt((x^3)^2+1/2+(x^(-3)/4)^2)=sqrt((x^3+x^(-3)/4)^2) =x^3+x^(-3)/4

Hence, the surface area can be expressed as:

S=2pi int_1^2 x sqrt(1+(dy/dx)^2) dx =2pi int_1^2(x^4+x^(-2)/4) dx

$= 2 \pi {\left[{x}^{5} / 5 - \frac{1}{4 x}\right]}_{1}^{2} = \frac{253 \pi}{20}$